I can generalize the notion of a space over $X$ (where $X$ is a based and connected space) to the notion of a spectrum over $X$ by considering functors of quasicategories $X\to Mod_\mathbb{S}$. In the language of recent work of Ando, Blumberg and Gepner such functors can be thought of as bundles (i.e. locally constant sheaves) of $\mathbb{S}$-modules over $X$, or as $X$-parameterized spectra. In the case that we have a space parameterized by $X$, we know relatively obviously that $\Omega X$ acts on the fiber over the base point. This of course comes from just thinking about the fact that we can let the fiber travel along the path traced out by an loop in $X$.

More generally however, it seems that it should be true that $\Omega X$ acts on the "fiber" of parameterized spectrum over $X$. How can I make a formal argument to this effect, which perhaps relies only on category-theoretic considerations? In particular, if I'm not mistaken, there is an equivalence between "spectra over $X$" and "spectra with an $\Omega X$-action" for which "taking the fiber" is one direction.

In general I feel like this should be very generally true for the category of functors $X\to \mathcal{C}$ for any (presentable?) quasicategory $\mathcal{C}$.

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    $\begingroup$ I think that the first occurrence of $X$ in the second paragraph was meant to be $\Omega X$. $\endgroup$ – Tom Goodwillie Jun 14 '15 at 16:36
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    $\begingroup$ $X$ needs to be a based and connected space. $\endgroup$ – Qiaochu Yuan Jun 14 '15 at 21:21
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    $\begingroup$ @QiaochuYuan that comment is not helpful at all. Obviously I understand that $X\simeq B\Omega X$. If I knew why that implied what I'm describing above, I wouldn't have asked the question. $\endgroup$ – Jonathan Beardsley Jun 16 '15 at 1:40
  • $\begingroup$ @Jon: the data of an action of a grouplike $E_1$ space $G$ on an object of an $\infty$-category $C$ is precisely the data of a functor $BG \to C$. So the data of a functor $X \to C$ is precisely the data of a functor $B \Omega X \to C$, which is in turn precisely the data of an $\Omega X$-action on an object of $C$. $\endgroup$ – Qiaochu Yuan Jun 16 '15 at 3:11
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    $\begingroup$ @QiaochuYuan It feels a bit like you are addressing the obvious part and not the interesting part of the question. I think Jon is aware of what you are saying, but I could be wrong. Heaven knows it has happened before. $\endgroup$ – Sean Tilson Jun 16 '15 at 14:04

I hope I've understood your question.

Any connected based space $X$ is weak homotopy equivalent to the classifying space of a topological group $G$ in a functorial way (the group is the realization of the Kan loop group of the simplicial total singular complex). So we can assume $X = BG$. Then there is an equivalence of homotopy categories $$ \text{ho(parametrized spectra}/X) \qquad \simeq \quad \text{ho(naive $G$-spectra)} $$ (This equivalence arises from a Quillen equivalence with respect to suitable choices of model structures.)

The functors inducing the equivalences are given from right to left by taking the unreduced Borel construction and from left to right by base change along $EG \to X$ and then collapsing out the zero-section.

Actually the situation above arises from a Quillen equivalence between the model categories "Spaces over X" and "$G$-spaces." Then one can pass to categories spectra on both sides to get the result.

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